Entropy as an Indication of the Runtime of Terminating Discrete Dynamical Systems
Résumé
Various hard problems in computer science are related to the asymptotical runtime of terminating processes.Most prominent among these problems is the question whether P = NP , that is, whether problems whosesolutions can be checked in polynomial time to be solutions can actually be solved in polynomial time. Inthis paper we make a first experimental study on how a certain notion of entropy can be an indication of theruntime of processes. To this extend we have programmed and ran all Turing machines with two symbols andeither two or three states. For a space-time representation of the tape-evolution of these machines we defineda notion of entropy. For all the machines with 2 states and 2 colors we have verified that the entropy is 1 if andonly if the process terminates in linear time, it is 2 if and only if it takes super-polynomial time and strictlybetween 1 and 2 for all asymptotic runtimes that are slower than linear but faster than super-polynomial.Ongoing work aims at testing whether this strong correlation can be extended to a more general setting.